conductivityTensor

Specify a tensor that has different conductivity parallel and perpendicular to a given vector field. The Tensor is specified as

given an input vector field \(B\) that same field is used to generate a unit vector field \(b\) that is then used to define the tensor. Conductivity parallel to the magnetic field is specified as \(K_{\parallel}\) and perpendicular to the vector field as \(K_{\perp}\) and then the difference in conductivities \(dK=K_{\parallel}-K_{\perp}\). The 9 tensor components are given as

\[\notag \begin{align} \left( \begin{array}{ccc} K_{\perp}+dK\,b_{x}^{2} & K_{\parallel}b_{x}b_{y} & K_{\parallel}b_{x}b_{z} \\ K_{\parallel}\,b_{x}b_{y} & K_{\perp}+dK\,b_{y}^{2} & K_{\parallel}b_{y}b_{z} \\ K_{\parallel}\,b_{x}b_{z} & K_{\parallel}b_{y}b_{z} & K_{\perp}+dK\,b_{z}^{2} \\ \end{array} \right) \end{align}\]

Parent Updater Data

in (string vector, required)

1st Variable

  1. \(V_{x}\) x vector component
  2. \(V_{y}\) y vector component
  3. \(V_{z}\) z vector component

2nd Variable

  1. \(K_{\parallel}\) parallel conductivity

3rd Variable

  1. \(K_{\perp}\) perpendicular conductivity
out (string vector, required)

The output variable is a length 9 vector containing the 9 components of the conductivity tensor

1st Variable

  1. \(T_{xx}\)
  2. \(T_{xy}\)
  3. \(T_{xz}\)
  4. \(T_{yx}\)
  5. \(T_{yy}\)
  6. \(T_{yz}\)
  7. \(T_{zx}\)
  8. \(T_{zy}\)
  9. \(T_{zz}\)

Example

 <Updater initConductivityTensor>
  kind = equation2d
  onGrid = domain

  in = [B, kParallel, kPerpendicular]

  out = [conductivityTensor]

  <Equation a>
    kind = conductivityTensor
  </Equation>
</Updater>